On non-displaceable Lagrangian tori on Fano toric surfaces: Wall-crossings and bulk-deformations

We adapt Lagrangian Floer theory on the de Rham complex developed by Fukaya [Fuk] and Fukaya-Oh-Ohta-Ono [FOOOToric2] and [FOOOToric3] for a Lagrangian torus fibration possibly with singular torus fibers under some assumptions, focusing on deformations of the Floer theory by degree 2 cycles from an ambient symplectic manifold. As an application, we detect non-displaceable torus fibers of a Lagrangian torus fibration with a singular fiber on Fano toric surfaces constructed by Auroux [Auroux1]. We detect a continuum of non-displaceable Lagrangian tori on some Fano toric surfaces that are not related to any standard toric fibers by any symplectomorphisms.